symmetry-adapted linear combinations

When several outer atoms surround a central one — six ligands around a metal, four hydrogens around carbon, three fluorines around boron — their orbitals are equivalent by symmetry, and you cannot sensibly say which single ligand orbital bonds with which central orbital. The trick is to first add the outer orbitals together in specific mixtures that match the molecule's symmetry. Those tailored mixtures are symmetry-adapted linear combinations, almost always shortened to SALCs.

Each SALC is built so that it transforms as one of the molecule's irreducible representations — it has a definite symmetry label like a1g or t1u. You find them by taking the orbitals of the surrounding atoms as a basis, generating their reducible representation, and reducing it to irreps; the irreps that come out tell you which symmetry types of combinations exist, and a related tool (the projection operator) gives the actual mixtures. The result is a set of group orbitals: in-phase and out-of-phase blends of the ligand orbitals, each carrying a clean symmetry tag.

SALCs are the indispensable bridge to a correct molecular-orbital or ligand-field diagram, because a central-atom orbital can only bond with a ligand combination that shares its symmetry label. So you match the metal's orbitals (s is a1g, the p set is t1u, the d set splits into eg plus t2g in Oh) to the ligand SALCs of the same labels, and the bonding and antibonding levels fall out by symmetry rather than guesswork. This is exactly how the molecular-orbital picture of an octahedral complex — and the more accurate ligand-field treatment that replaces the point-charge crystal-field model — is actually constructed.